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Theorem rmo5 2646
Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
Assertion
Ref Expression
rmo5 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))

Proof of Theorem rmo5
StepHypRef Expression
1 df-mo 2003 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) → ∃!𝑥(𝑥𝐴𝜑)))
2 df-rmo 2424 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
3 df-rex 2422 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
4 df-reu 2423 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
53, 4imbi12i 238 . 2 ((∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) → ∃!𝑥(𝑥𝐴𝜑)))
61, 2, 53bitr4i 211 1 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wex 1468  wcel 1480  ∃!weu 1999  ∃*wmo 2000  wrex 2417  ∃!wreu 2418  ∃*wrmo 2419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116  df-mo 2003  df-rex 2422  df-reu 2423  df-rmo 2424
This theorem is referenced by:  nrexrmo  2647  cbvrmo  2653  bdrmo  13138
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